STINGY SAT is the following problem: given a set of clauses...

STINGY SAT is the following problem: given a set of clauses (each a disjunction of literals) and an integer $k,$ find a satisfying assignment in which at most $k$ variables are true, if such an assignment exists. Prove that STINGY SAT is NP-complete.

Key Concepts

NP, or Non-deterministic Polynomial time, is a class of decision problems for which a given candidate solution can be verified in polynomial time. This concept is fundamental in theoretical computer science, particularly in discussions of computational complexity, as it sets the stage for understanding the difficulty of problems and the efficiency of algorithms that solve or verify them.

NP-completeness

NP-completeness refers to a subset of NP problems that are as hard as any other problem in NP. A problem is NP-complete if every problem in NP can be reduced to it in polynomial time and it itself belongs to NP. This concept is crucial because if any NP-complete problem can be solved in polynomial time, then all problems in NP can be solved in polynomial time.

Boolean Satisfiability (SAT)

The Boolean Satisfiability Problem (SAT) is the problem of determining if there exists an assignment of truth values to variables such that a given Boolean formula evaluates to true. SAT is the first problem that was proven NP-complete and serves as a starting point for reductions to demonstrate the NP-completeness of numerous other computational problems.

Polynomial Time Reductions

Polynomial time reductions are transformations from one problem to another that can be performed in polynomial time. These reductions are used to show that if one problem is NP-hard, then another problem is also NP-hard by transferring the inherent difficulty through the reduction. This concept is pivotal in complexity theory for proving NP-completeness.

Cardinality Constraints in Constraint Satisfaction

Cardinality constraints in constraint satisfaction problems refer to restrictions on the number of variables that can be assigned a particular value, such as limiting the number of variables that can be true. This additional layer of constraint adds complexity to the problem and requires careful handling in proofs and reductions to demonstrate NP-completeness.

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